An improved estimate for the Bloch norm of functions in Doob’s class

Abstract

For any fixed 0 > ρ > 2 π , D ( ρ ) 0 > \rho > 2\pi ,\mathcal {D}(\rho ) is the family of all holomorphic functions in Δ \Delta which satisfy (i) f ( 0 ) = 0 f(0) = 0 , and (ii) lim _ z → τ | f ( z ) | ⩾ 1 {\underline {\lim } _{z \to \tau }}|f(z)| \geqslant 1 , for all τ \tau lying on some arc Γ f ⊆ ∂ Δ {\Gamma _f} \subseteq \partial \Delta with arclength | Γ f | ⩾ ρ |{\Gamma _f}| \geqslant \rho . We showed that for each f ∈ D ( ρ ) f \in \mathcal {D}(\rho ) there exists a point z f ∈ Δ {z_f} \in \Delta at which \[ | f ′ ( z f ) | ( 1 − | z f | 2 ) ⩾ 2 e sin ⁡ ( π − ( ρ / 2 ) ) ( π − ( ρ / 2 ) ) . |f’({z_f})|(1 - |{z_f}{|^2}) \geqslant \frac {2}{e}\frac {{\sin (\pi - (\rho /2))}}{{(\pi - (\rho /2))}}. \] In this paper we improve this estimate by replacing the quantity π − ( ρ / 2 ) \pi - (\rho /2) with a value θ ( ρ ) \theta (\rho ) which lies between 0 and π − ( ρ / 2 ) \pi - (\rho /2) and so improves the estimate. The value θ ( ρ ) \theta (\rho ) is defined as the (unique) solution in this interval of the equation F ρ ( θ ) = log ⁡ ( cot ⁡ ( ρ / 4 ) cot ⁡ ( θ / 2 ) ) − θ / sin ⁡ θ = 0. {F_\rho }(\theta ) = \log (\cot (\rho /4)\cot (\theta /2)) - \theta /\sin \theta = 0. .